Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to solve the inequality. (Round each number in your answer to two decimal places.) \(-0.5 x^{2}+12.5 x+1.6>0\)

Step-by-Step Solution

Answer

The solution to the inequality are the intervals which make the inequality true when tested. Remember to round the interval limits to two decimal places.

Step 1: Formulate the quadratic equation

Firstly, rewrite the inequality into the quadratic equation: \(-0.5 x^{2}+12.5 x+1.6=0\).

Step 2: Solve the quadratic equation

Use a calculator to find the roots of the quadratic equation by using the quadratic formula \(x = [-b ± sqrt(b^2 - 4ac)]/(2a)\), where \(a = -0.5\), \(b = 12.5\) and \(c = 1.6\). This will give two roots, 'x1' and 'x2', which are the points where the inequality changes its sign.

Step 3: Determine the intervals

With the obtained roots, we can split the number line into three intervals: \(-∞ < x < x1\), \(x1 < x < x2\) and \(x2 < x < ∞\). Pick a test point in each interval and plug it into the inequality to see if the statement is true.

Step 4: Solve and test for inequality

For the test points in intervals where the inequality is true, include that interval as part of the solution. If it's false, exclude the interval. Remember to round to two decimal places as required.

Key Concepts

Quadratic Equation

In mathematics, a quadratic equation is an equation of the second degree, meaning it includes at least one term that is squared. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(a \eq 0\).

In the given exercise, the quadratic inequality \( -0.5 x^{2}+12.5 x+1.6>0 \) can be formulated as a quadratic equation by setting it equal to zero, which is the first crucial step in finding its roots. The equation \( -0.5 x^{2}+12.5 x+1.6=0 \) is necessary to determine where the function crosses the x-axis, which then helps identify the intervals for the inequality's solutions.

Quadratic Formula

The quadratic formula is a pivotal tool for solving quadratic equations. It provides the solutions for \(x\) by applying the formula \(x = [-b \pm \sqrt{b^2 - 4ac}]/(2a)\). This is derived from completing the square in the standard form of a quadratic equation. In our exercise, we use the coefficients \(a = -0.5\), \(b = 12.5\), and \(c = 1.6\) to find the roots of the equation. By substituting these values into the formula, we determine the critical points where the inequality will change its direction, which are essential for pinpointing the range of values for \(x\) that satisfy the original inequality.

Inequality Solution

To solve an inequality means to find all of its solutions, which are the values that make the inequality true. Unlike equations, inequalities do not have just one solution but a range of possible solutions. After finding the roots of the quadratic equation, we split the number line into intervals based on these roots. Then, we test points from each interval in the original inequality. For instance, if our test point when plugged into the inequality yields a true statement, then this interval is part of the solution set. As in the exercise provided, one must carefully test and verify these intervals with suitable test points and include only those intervals where the inequality holds true.

Graphical Representation of Inequalities

Graphically, inequalities are represented by shading the regions where the inequality is satisfied. For a quadratic inequality, the graph is a parabola, and the shaded area indicates where the inequality is true. After solving the corresponding equation, the roots denote where the parabola intersects with the x-axis. For \( -0.5 x^{2}+12.5 x+1.6 > 0 \), we'd see the direction of the parabola and the intervals above the x-axis that satisfy the inequality. To further aid understanding, a number line is used to visualize these intervals, often with a solid circle at the points if they are included in the solution or open circles if not.

Interval Notation

Interval notation is a way of writing subsets of the real numbers. This notation is incredibly efficient for expressing the solution sets of inequalities. It uses brackets to indicate closed intervals where endpoints are included, and parentheses for open intervals where endpoints are not included. For example, if \(x1 \eq x2\) are the roots of the quadratic equation, the solution in interval notation might look something like \( (x1, x2) \) or \( [x1, x2] \) depending on whether the end values are part of the solution. It is a concise way to communicate the range of solutions and is a valuable tool for students to understand when solving inequalities like the one presented in the exercise.